General Considerations

The question of whether a positron can bind itself to a neutral atom and form an electronically stable bound state has been one of the longest standing questions in positron physics [7, 8] with a number of negative or inconclusive results [9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. In the first instance, the static interaction between the atom and the positron is repulsive everywhere, and therefore at first sight represents an environment which is inimical to binding a positron.

However, the electronic charge cloud of the atom, can adjust itself to accommodate the presence of a nearby positron. The polarization of the electron charge cloud leads to an attractive interaction between the positron and the atom. The polarization potential is known to have the asymptotic form where ay is the static dipole polari/ahilily. All of the equations in this chapter are given in atomic units for which ao = 1, e = 1, me = 1 and h = 1. A bound state will occur when the attractive polarization potential is large enough to overcome the repulsive interaction with the nucleus.

There is another possible binding mechanism. In circumstances where the ionization potential of the atom is less than 0.250 Hartree (the Ps binding energy) it is possible for one of the valence electrons to attach itself to the positron forming a Ps cluster. The polarization of this Ps cluster by the Coulomb field of the residual singly charged ion core results in an attractive interaction that can also lead to binding.

2.1. Condition for Positron Binding

The condition for positron binding to an atom has a subtle dependence upon the ionization potential, I, of the atom. If I is greater than 0.250 Hartree, then the question of positron binding is just a question of whether the atom has a positron affinity, i.e. whether the ground state energy of the e+A system is less than the ground state energy of the neutral atom. However, if the ionization energy is less than 0.250 Hartree, the binding energy of the positron to the atom must exceed Hartree otherwise the positron-atom complex will dissociate into positronium plus a residual positively charged ion, i.e. into Ps + A+. When the parent atom has an I < 0.250 Hartree one must establish that the PsA+ system is bound.

The binding energy of the system is denned as the binding energy with respect to the lowest energy dissociation channel. The positron affinity is defined as the binding energy gained by the positron when it is attached to the atom. The binding energy and positron affinity are only equal for atoms with I > 0.250 Hartree.

2.2. Positron Annihilation

Positron annihilation is a simple process. Whenever an electron and positron come into direct contact they will annihilate. If the spin state of the annihilating pair is a singlet (S = 0) state, the dominant decay process is the 27 decay. In the spin-triplet state the dominant process is 37 decay. For the simplest of all positron binding systems, positronium, the singlet state decays at a rate equal to 47rr§c/(87raj|) = 8.0325 x 109 sec-1. Triplet Ps decays at a rate of ^(tt2 - 9)mec?a6/h = 7.211 x 106 sec-1 [19] (higher order terms in a act to reduce both of these rates slightly). In these expressions, a is the fine-structure constant, r<> is the classical electron radius and c is the speed of light.

Complications arise when a positron bound to a complex electronic system decays. The final state will consist of a residual ion in a specific quantum state with a recoil momentum that depends on the center of mass momentum of the annihilating electron-positron pair [20, 21, 22, 23, 24], The net 27 annihilation rate of a positronic atom with wave function ^(ri,..., rjv; ro) with N electrons resulting in the emission of two gamma quanta with total momentum q is defined as

The second rjv co-ordinate in eq.(2) denotes the positron. Since i is anti-symmetric for electron interchange, the index of the annihilating electrons is sel as i = N. T is a constant defined as

where rois the classical electron radius. The operator, Ôf in eq.(2) is a spin projection operator to the singlet state of the positron-(ith electron) pair, which can be written as

Equation (2) can be integrated over q to give where r is used to denote the complete electron phase space d3r = nJLidV The expression for T is the expression that is commonly called the 27 annihilation rate in the literature. Equation (5) does not give the transition rate between a well defined initial and final state. This equation is a sum rule which adds up the individual transition rates over all possible final states. All the annihilation rates presented in this article are spin-averaged and only take into consideration the 27 process. The spin-averaged 27 annihilation rate for the Ps ground state is 2.008 x 109 s-1.

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